and Cosmology

7.5 Non-Linear Structure Evolution
Combining (7.37) and (7.38), we can express the halo
mass and virial radius as a function of the virial velocity,
M = V 3 200
10GH(z) , r 200 = V 200
10H(z) . (7.39)
Since the Hubble function H(z) increases with redshift,
the virial radius at fixed virial velocity decreases with
redshift. From (7.37) we also see that r 200 decreases
with redshift at fixed halo mass. Hence, halos at a given
mass (or given virial velocity) are more compact at
higher redshift than they are today.
The NFW Profile. The density profile of halos averaged
over spherical shells seems to have a universal functional
form, which was first reported by Julio Navarro,
Carlos Frenk & Simon White in a series of articles in
the mid-1990s. This NFW-profile is described by
ρ(r) =
ρ s
(r/r s )(1 +r/r s ) 2 , (7.40)
where ρ s is the amplitude of the density profile, and
r s specifies a characteristic radius. For r ≪ r s we find
ρ ∝ r −1 , whereas for r ≫ r s , the profile follows ρ ∝ r −3 .
Therefore, r s is the radius at which the slope of the density
profile changes (see Fig. 7.13). ρ s can be expressed
in terms of r s , since, according to the definition of r 200 ,
ρ = 200ρ cr (z) = 3
4πr 3 200
∫1
= 3ρ s
0
∫
r 200
0
4πr 2 dr ρ(r)
dx x 2
cx(1 + cx) 2 , (7.41)
where in the last step the integration variable was
changed to x = r/r 200 , and the concentration index
c := r 200
r s
(7.42)
was defined. The larger the value of c, the more strongly
the mass is concentrated towards the inner regions.
299
Fig. 7.13. For eight different cosmological
simulations, the density profile is shown for
the most massive and the least massive halo,
each as a function of the radius, together
with the best fitting density profile (7.40).
The cosmological models represent an EdS
model (here denoted by SCDM), a ΛCDM
model, and different models with power
spectra that are assumed to be power laws
locally, P(k) ∝ k n . The arrows indicate the
softening length in the gravitational force
for the respective halos; thus, the major part
of the profiles is numerically well resolved